Neuro-ODE in Modeling Hub
Data and Code are avaiable: here
Project Framework
Baseline Model
SIR Model
In our initial calibration stage, we first focus on simple SIR-based model following the 19 round from COVID-19 Scenario Modeling Hub. β was estimated using least square method.
In the model, susceptible individuals (S) move to the vaccinated compartment (V) through vaccination or, upon infection, transition to either a not-immunized hospital-bound infectious compartment (Inh) or a not-immunized non-hospitalized infectious compartment (Inr). Vaccinated individuals can still become infected and transition to either a immunized hospital-bound infectious compartment (Imh) or a immunized non-hospitalized infectious compartment (Imr). All hospital-bound infections eventually transition to either the recovering hospitalized compartment (Hr) or the fatal hospitalized compartment (Hd). Non-hospitalized infectious individuals and recovering hospitalized patients eventually transition to the recovered compartment (R), while fatal hospitalizations ultimately transition to the death compartment (D). Recovered individuals gradually lose immunity and become partially susceptible (S*); upon reinfection, they may transition to the immunized infectious compartment (Imr or Imr), or return fully to the susceptible compartment (S). yvac,t is the observed vaccinated population at time t.
Data
| Data | Description | Ref |
|---|---|---|
| Hospitalization | Count of state-level hospitalizations over time | Link |
| Vaccination Coverage | survey-based historical coverage | Link |
Data from Texas were used (state code: 48) for all age groups. Hospitalization data span November 9, 2024 to April 26, 2025, covering a complete epidemic wave. The corresponding vaccination data span November 10, 2024 to December 29, 2024, with the vaccination window shifted forward by one day to align with the modeling timeline. Figures as below:
Model Parameters
1. Vaccine-related parameters
The full source is available here.
| Parameter | Value | Description | Ref |
|---|---|---|---|
| Vaccine Efficacy against Infection | 35-57% | CDC_VE | |
| Vaccine Efficacy against Hospitalization | 45% | CDC_MMWR | |
| Minimun Time of Reinfection | 19 days | Reinfection notates &rho from R to S* | Tang,2021 |
| Waning Time | 3-10 months | Waning notates ω from S* to S | NIH |
| Vaccine Efficacy Reduction | 40-60% | Reduction against baseline protection levels in waned state | Andrews,2022 |
| Immune Escape | 20-50% | Protection against current variants with the antibodies from the previous variant is X * 0.8 (20% immune escape) | Chen,2024 |
*Blue color represents parameters that have not yet been used.
2. Other Model Parameters
| Parameter | Value | Ref |
|---|---|---|
| Total Population (N) | 31,290,831 | US Census |
| Latent rate (σ) | 1/1~1/3 | He,2020 |
| Recovery rate from Infectious (γ1) | 1/5~1/7 | CDC_COVID |
| Recovery rate from Hospitalized (γ2) | 1/7~1/14 | Rees,2020 |
| Infectious to Hospitalization rate (IHR) | 0.01~0.02 | Deng, 2025 |
| Time from Infectious to Hospitalization (+latent) (δ) | 8~10 days | Linton,2020 |
| Hospitalization to Death rate (HDR) | 0.16 | Baptista,2023 |
| Time from Hospitalization to Death (η) | 10~14 days | Verity,2020 |
| Reporting rate (rep) | 0.2 | Smirnova,2024 |
| Reporting delay | 1 week | Assume |
*Blue color represents parameters that have not yet been used.
3. Initialize Compartments
| Parameter | Value | Ref |
|---|---|---|
| Initial Vaccinated (V0) | 2,606,590 | Cumulative vaccination before November 9, 2024 (from Aug 2024) |
| Initial Not-immunized Infectious (In,0) | 160,000 | Assume |
| Initial Immunized Infectious (Im,0) | 80,000 | Assume |
| Initial Recovered (R0) | 200,000 | Assume |
| Initial Hospitalized (H0) | 10,000 | Assume |
| Initial Death (D0) | 70 | Assume |
| Initial Partially Susceptible (S0*) | 6,000,000 | Assume |
Model Fitting
Fitted β = 0.246
Model Projection
1. Two-years Projection with Stochasticity
We did not consider new vaccination and new variants after 2025-04-26 in projection.
To represent short-term fluctuations in transmission while avoiding unrealistic long-term drift, we model the time-varying transmission rate βt as a mean-reverting Itô stochastic process (Ornstein–Uhlenbeck, OU). Intuitively, βt can randomly go up or down from day to day (dt = 1 (day)), but it is gradually pulled back toward a typical level β̄.
Here, Wt is standard Brownian motion. Over a time step dt, the Brownian increment satisfies ΔWt = Wt+dt − Wt ~ N(0, dt), which can be simulated as ΔWt = √dt · Zt with Zt ~ N(0, 1).
We approximate the OU process using the Euler–Maruyama method. The discrete-time update is:
Parameter interpretations:
- β̄ is the long-run average transmission level and corresponds to the transmission rate estimated from the deterministic model fit.
- κ controls how quickly βt returns toward β̄ (larger κ = faster reversion), derived from autocorrelation structure of log-scale model residuals.
- σ controls the magnitude of short-term randomness (larger σ = noisier day-to-day changes), derived from standard deviation of log-scale model residuals.
*Grey vertical line represents the cut-off point, data before the cut-off point is used to fitting.
2. Sensitive Analysis
Next Step
1. Age-specific transmission/vaccination/IHR...
2. Neuro-ODE